Is there a shortcut or hotkey that I could use to access the Transform panel of the properties shelf in the 3D viewport?
You can also use the keyboard shortcuts combined with the Arrow Keys.

For example: in top view pressing G and the up/down arrow keys will move the selected vertices on the vertical (Y) axis. The left and right arrows will move on the horixontal (X) axis.

On any side view, the up/down arrows will control movement on the Z axis.

You can still use constraints like GX, G Y, GZ to limit the movement to one axis

The precision of your movements will depend on how far you zoom in on the viewport.

For even smaller moves you can press Shift while pressing the arrow keys.